Properties of Norm on Division Ring/Norm of Quotient

Theorem

Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.

Let $\norm {\,\cdot\,}$ be a norm on $R$.

Let $x, y \in R$

Then:

$y \ne 0_R \implies \norm {x y^{-1} } = \norm {y^{-1} x} = \dfrac {\norm x} {\norm y}$

Proof

Let $y \ne 0_R$.

By Norm Axiom $\text N 1$: Positive Definiteness then:

$\norm y \ne 0$

So:

\(\ds \norm {x \circ y^{-1} }\)

\(=\)

\(\ds \norm x \norm {y^{-1} }\)

Norm Axiom $\text N 2$: Multiplicativity

\(\ds \)

\(=\)

\(\ds \dfrac {\norm x} {\norm y}\)

Norm of Inverse

Similarly:

$\norm {y^{-1} x} = \dfrac {\norm x} {\norm y}$

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields, Theorem $1.6 \ \text {(e)}$